
In the second estimate we used that \(|x\ln (x)| \le C_{p}\,x^{\frac{1}{r}}\) , for any \(r>1\)  and \(x\in [0,1]\) . Furthermore, it is well known that using energy conservation and entropy propagation it follows for the solution of the homogeneous Boltzmann equation \(f(v,t)\) , see [1]} or more recently [2]}, that
\(\sup _{t\ge 0}\int _{\mathbb {R}^{d}}f(v,t)\big |\ln \big (f(v,t)\big )\big | dv \le C\bigg (\int f_0\ln (f_0)dv,\int f_0| v |^{2}dv\bigg )=:C_{\mathcal {M}}\,.\) 
